Decision Forests - Data Mining with Decision Trees: Theory and Applications - page 111

where:

P M k ( y = c j |

x )log( P M k ( y = c j |

E ( M k ,x )= − x ) . ) (9.12)

c j

9.3.1.9 Density-based Weighting

If the various classifiers were trained using datasets obtained from different
regions of the instance space, it might be useful to weight the classifiers
according to the probability of sampling x by classifier M k ,namely:

P M k ( x ) .

Class ( x ) = argmax
c i ∈dom ( y )

(9.13)

P M k ( y = c j |x )

k : c i =argmax
c j ∈ dom ( y )

The estimation of P M k ( x ) depends on the classifier representation and can
not always be estimated.

9.3.1.10 DEA Weighting Method

Recently there has been attempt to use the data envelop analysis (DEA)
methodology [ Charnes et al . (1978) ] in order to assign weights to different
classifiers [ Sohn and Choi (2001) ] . These researchers argue that the weights
should not be specified according to a single performance measure, but
should be based on several performance measures. Because there is a trade-
off among the various performance measures, the DEA is employed in
order to figure out the set of ecient classifiers. In addition, DEA provides
inecient classifiers with the benchmarking point.

9.3.1.11 Logarithmic Opinion Pool

According to the logarithmic opinion pool [Hansen (2000)], the selection of
the preferred class is performed according to:

e P

α k · log( P M k ( y = c j |x )) ,

Class ( x ) = argmax
c j ∈dom ( y )

(9.14)

k

where α k denotes the weight of the k -th classifier, such that:

0; α k =1 . α k ≥ (9.15)